\newproblem{lay:2_1_23}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 2.1.23}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
  Let $A$ be an $m\times n$ matrix. Suppose there exists an $n\times m$ matrix $C$ such that $CA=I_n$ (the $n\times n$ identity matrix).
	Show that the equation $A\mathbf{x}=\mathbf{0}$ has only the trivial solution. Explain
	why $A$ cannot have more columns than rows.
}{
  % Solution
	If $\mathbf{x}$ satisfies $A\mathbf{x}=\mathbf{0}$, then
	\begin{center}
		$CA\mathbf{x}=C(A\mathbf{x})=C\mathbf{0}=\mathbf{0}$.
	\end{center}
	But on the other side
	\begin{center}
		$CA\mathbf{x}=(CA)\mathbf{x}=I_n\mathbf{x}=\mathbf{x}$.
	\end{center}
	Consequently, $\mathbf{x}=\mathbf{0}$. This shows that the equation $A\mathbf{x}=\mathbf{0}$ has no free variables. A requirement for this is that there
	are not more columns than rows.
}
\useproblem{lay:2_1_23}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
